报 告 人：冯衍全 教授

报告题目：Semiregular and quasi-semiregular automorphisms of digraphs

报告时间：2024年12月15日（周日）下午3：00

报告地点：静远楼1508会议室

主办单位：数学与统计学院、数学研究院、科学技术研究院

报告人简介：

       冯衍全，北京交通大学二级教授，自1997年获北京大学理学博士学位以来，一直从事代数与组合，群与图以及互连网络方面研究。现任中国工业与应用数学学会理事、中国数学会理事等，代数组合JACO等杂志编委。2010年主持《图的对称性》获教育部优秀成果二等奖，2011年获政府特殊津贴。共发表SCI科研论文150余篇，主持完成国家自然科学基金10余项，包括重点项目1项。正在承担国家自然科学基金重点项目1项、面上项目1项、国际合作研究项目1项。

报告摘要：

       Let G be a permutation group on a finite set Omega . An non-identity element g in G is said to be semiregular if every cycle in the unique cycle decomposition of g has the same length, and quasi-semiregular if g has an unique 1-cycle in the cycle decomposition of g and every other cycle has the same length. An automorphism of a digraph is called semiregular or quasi-semiregular if it is a semiregular or quasi-semiregular permutation on the vertex set of the digraph. The permutation group G is called 2-closed if G is the largest subgroup of the symmetric group S_Omega on Omega with the same orbits as G on Omega× Omega.

       In 1981 Fein, Kantor and Schacher proved that a transitive permutation group on a finite set with degree at least 2 has an element of prime-power order with no fixed point, but may not have a semiregular element. In the same year, Marusic conjectured that every finite vertex-transitive digraph has a semiregular automorphism, and in 1995, Klin proposed the well-known Polycirculant Conjecture: Every 2-closed transitive permutation group has a semiregular element. Note that the automorphism group of any digraph is 2-closed. In 2013, Kutnar, Malnic, Martanez and Marusic proposed the quasi-semiregular automorphism of a digraph and investigated strongly regular graphs with such an automorphism.

        A lot of work relative to semiregular or quasisemiregular automorphisms of digraphs has been done and in this talk, we review some progress on this line. Furthermore, we talk about a recent work by Yin, Feng, Zhou and Jia [Journal of Combinatorial Theory B 159 (2023) 101-125] on prime-valent symmetric graphs with a quasi-semiregular automorphism.